> ## Documentation Index
> Fetch the complete documentation index at: https://cyberpaisa-dof-mesh-40-27.mintlify.site/llms.txt
> Use this file to discover all available pages before exploring further.

# Z3 Verifier API

> Z3Verifier, TransitionVerifier, HierarchyZ3 — ProofResult states, invariants, and hierarchy patterns.

## Z3Verifier

**File:** `core/z3_verifier.py`

```python theme={null}
from dof import Z3Verifier

verifier = Z3Verifier()

# Run all 4 theorems
results = verifier.verify_all()
# [ProofResult(name="GCR_INVARIANT", status="VERIFIED", time_ms=3.10), ...]

# Verify governance state
result = verifier.verify_governance_state(
    gcr=0.95,
    supervisor_score=0.87,
    previous_score=0.81
)
print(result.verdict)     # "APPROVED"
print(result.proof_hash)  # "0x7f3a9c..."
```

***

## ProofResult States

| Status                 | Z3 equivalent | Meaning                                                  |
| ---------------------- | ------------- | -------------------------------------------------------- |
| `VERIFIED`             | UNSAT         | Property holds universally — no counterexample exists    |
| `COUNTEREXAMPLE_FOUND` | SAT           | Counterexample found — property violated                 |
| `TIMEOUT`              | —             | Solver exceeded time budget                              |
| `PROVEN`               | UNSAT         | `TransitionVerifier` equivalent of VERIFIED              |
| `VIOLATED`             | SAT           | `TransitionVerifier` equivalent of COUNTEREXAMPLE\_FOUND |

***

## The 4 Theorems

```python theme={null}
from dof import Z3Verifier

v = Z3Verifier()
results = v.verify_all()

for r in results:
    print(f"{r.name}: {r.status} ({r.time_ms}ms)")
# GCR_INVARIANT:     VERIFIED (3.10ms)
# SS_FORMULA:        VERIFIED (1.80ms)
# SS_MONOTONICITY:   VERIFIED (2.40ms)
# SS_BOUNDARIES:     VERIFIED (1.20ms)
```

***

## TransitionVerifier (Z3Gate)

**File:** `core/z3_gate.py`

Verifies 8 state invariants across 9 transition types.

```python theme={null}
from core.z3_gate import Z3Gate

gate = Z3Gate()
result = gate.verify_transition(
    transition_type="PUBLISH",
    state={"trust_score": 0.85, "threat_detected": False}
)
print(result.verdict)  # "APPROVED" | "REJECTED" | "TIMEOUT" | "FALLBACK"
```

### 9 Transition Types

```
PUBLISH · SCORE_UPDATE · PROMOTE · DEMOTE · THREAT_DETECT
THREAT_CLEAR · COOLDOWN_START · COOLDOWN_END · GOVERNOR_ACTION
```

### 8 State Invariants

| ID    | Invariant                                                         |
| ----- | ----------------------------------------------------------------- |
| INV-1 | `threat_detected` → NOT `publish_allowed`                         |
| INV-2 | `trust_score < 0.4` → `attestation_count == 0`                    |
| INV-3 | `hierarchy_level_next <= hierarchy_level + 1`                     |
| INV-4 | `0 <= trust_score <= 1`                                           |
| INV-5 | `cooldown_active` → NOT `publish_allowed`                         |
| INV-6 | `hierarchy_level == 3` → `trust_score > 0.8`                      |
| INV-7 | `safety_score == 1 - failure_rate³`                               |
| INV-8 | `governance_violation` → `hierarchy_level_next < hierarchy_level` |

***

## HierarchyZ3

**File:** `core/hierarchy_z3.py`

```python theme={null}
from core.hierarchy_z3 import HierarchyZ3

hz3 = HierarchyZ3()

# Verify all 42 patterns
result = hz3.verify_hierarchy_inviolable()
print(result)  # "PROVEN"

# Check for weak patterns
weak = hz3.find_weakest_pattern()
print(weak)  # None  (all strong)

# Pattern breakdown
print(hz3.override_patterns)    # 6
print(hz3.escalation_patterns)  # 11
print(hz3.total_patterns)       # 42
```

***

## quick.prove

```python theme={null}
from dof.quick import prove

result = prove()

assert result["verified"] == True
assert len(result["theorems"]) >= 4

for t in result["theorems"]:
    assert t["result"] == "VERIFIED"
    print(f"{t['name']}: {t['time_ms']}ms")
```

***

<CardGroup cols={2}>
  <Card title="Formal Verification" icon="function" href="/concepts/formal-verification">
    Theory, proof chain, on-chain receipts
  </Card>

  <Card title="CLI: prove" icon="terminal" href="/reference/cli-prove">
    dof prove, verify-states, verify-hierarchy
  </Card>
</CardGroup>